In this mission, can a player capture a control point already controlled by the other player? If so, does it overwrite or overlap?

You can put your token and score your points regardless if there is an opponents token on it or not. This mission's rules don't say anything about your token leaving the point when an opponent puts his token on the same point, so your token remains, therefore by the mission rules you aren't allowed to put a token on a point which already has your token on it, also regardless if there is an opponents token on it.